An approximation algorithm for alphabet indexing problem

Abstract

A K-indexing is a mapping from an alphabet Σ to a set Γ of K symbols forming the homomorphism to transform strings. Given Σ, two disjoint sets of strings P, Q over Σ and Γ={1,..., K}, Alphabet Indexing is the problem to find a K-indexing that transforms no two different strings taken from each P and Q into the same one. Although this problem is NP-complete, applying K-indexing to input data brings remarkable advantages in actual applications. In this paper, we introduce Max K-Indexing, a maximization version of Alphabet Indexing, that intends to maximize the number of pairs in P×Q whose strings are not transformed into the same ones. We show that the problem is MAX SNP-hard, that is, the problem seems to have no polynomial-time algorithm achieving an arbitrary small error ratio. Then we propose a simple polynomial-time greedy algorithm and show that the algorithm attains the constant error ratio 1/K for K-indexing. Also we define M-Tuple K-Indexing problem by extending pairs of strings in Max K-Indexing to tuples of strings over more than two sets, and show that a natural extension of the algorithm also achieves a constant error bound.